Question 5 of 7: RC-Ladder Phase-Shift Oscillator (Virtual-Ground Feedback)

Question 5: RC-Ladder Phase-Shift Oscillator (Virtual-Ground Feedback) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three identical series-$C$/shunt-$R$ sections ($V_C(=V_o)\to C\to V_B\to R\to$gnd$\to C\to V_A\to R\to$gnd$\to C\to$ virtual ground), plus $R_F$ feeding back directly from $V_o$ to the same inverting-input node; $R=10\text{k}\Omega$, $C=15\text{nF}$.

Find. $V_A(V_C)$, $V_B(V_C)$; the loop-gain expression $V_o/V_C$; the frequency and $R_F$ value for sustained oscillation.

Approach. $V_C$ and $V_o$ are the same physical node (redrawn on opposite sides of the schematic), but treating $V_C$ as an independent test input — standard practice for deriving a Barkhausen loop gain — lets the ladder's transfer function to the virtual-ground input be found by ordinary nodal analysis, then $R_F$ closes the loop by supplying whatever current the third capacitor demands at the (zero-current) inverting input. Barkhausen's criterion (loop gain $=+1$, purely real) then fixes both the oscillation frequency and the required $R_F$.

1. Part (a) — $V_A$, $V_B$. With $x\equiv1/(sRC)$ (dimensionless), nodal KCL at $V_B$ and $V_A$ (inverting input held at the ideal virtual ground, $0\text{V}$) gives, after eliminating $V_B$: $$\boxed{V_A=\dfrac{V_C}{(x+1)(x+3)}},\qquad\boxed{V_B=V_A(x+2)=\dfrac{V_C(x+2)}{(x+1)(x+3)}}$$ where $x=1/(sRC)$; this factors from the same $(x+2)^2-1=(x+1)(x+3)$ combination that the 2-shunt-node ladder algebra always produces.
2. Part (b) — loop gain $V_o/V_C$. The inverting input draws no current, so the current arriving through the third capacitor ($V_A\cdot sC$) must return entirely through $R_F$ from $V_o$: $V_A\,sC+V_o/R_F=0\Rightarrow V_o=-V_A\,sC\,R_F$. Substituting $V_A$ from Part (a) and simplifying ($sC=1/(Rx)$): $$\frac{V_o}{V_C}=\boxed{\dfrac{-R_F/R}{x\big[(x+1)(x+3)\big]}}=\dfrac{-C^3R^2R_F\,s^3}{3C^2R^2s^2+4CRs+1}$$ (the two forms are algebraically identical.)
3. Part (c) — oscillation frequency. Setting $s=j\omega$ and writing $y=\omega RC$, carrying the full $s^3=-j\omega^3$ factor from the numerator through $x(x+1)(x+3)$ leaves the loop gain's real part multiplied by $\big[(1-3y^2)+j4y\big]$ in the denominator. The loop gain is purely real (Barkhausen's phase condition) exactly when this bracket's imaginary part vanishes: $$1-3y^2=0\;\Longrightarrow\;y=\omega RC=\frac{1}{\sqrt3}$$ $$f_0=\frac{\omega_0}{2\pi}=\frac{1}{2\pi\sqrt3\,RC}=\frac{1}{2\pi\sqrt3(10\text{k}\Omega)(15\text{nF})}=\boxed{613\text{ Hz}}$$
4. Part (d) — $R_F$ for oscillation. Evaluating the (now purely real) loop gain at $\omega_0$ gives $V_o/V_C=R_F/(12R)$ (the numeric denominator collapses to exactly $12$ at $y=1/\sqrt3$). Barkhausen's magnitude condition (loop gain $=+1$ for sustained oscillation) then requires $$\frac{R_F}{12R}=1\;\Longrightarrow\;R_F=12R=12(10\text{k}\Omega)=\boxed{120\text{ k}\Omega}$$ (In practice $R_F$ is set a few percent above $120\text{k}\Omega$ to guarantee start-up from noise, with amplitude ultimately capped by op-amp saturation or an added limiter, exactly as in any linear-feedback oscillator.)